28) Which of the following imply that a function f is continuous at x = a?
a) For every ε > 0, there exists a δ > 0 such that if |x - a| < δ, then |f(x) - f(a)| < ε.
b) For every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - f(a)| < ε.
c) For every sequence {rn} with lim Tn = a, we have lim f(rn) = f(a).
d) All of the above.
29) In applying the ε - definition of continuity to the function f(x) = 8x - 5 at x = 1, given ε > 0, we can choose
a) δ = ε
b) δ = g
c) δ = ε/8
d) None of the above
30) In applying the ε - definition for lim for the function f(x) = x^2, given 0 < ε, we can assume that 0 < δ < 1, so that if 0 < |x - 3| < δ then 2 < x < 4 and we can choose
a) δ = min{1, ε} since δ < 1.
b) δ = min{1, δ} since x < 4
c) δ = min{1, δ} since |x + 3| < 7.
d) None of the above.
